DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Irreducible, completely reducible, and faithful representations

Definition

A nonzero representation $V$ is irreducible if its only stable linear subspaces are $0$ and $V$ .

A representation is completely reducible if it is isomorphic, as a representation, to an algebraic direct sum

$⨁_{i \in I} V_{i}$

of irreducible representations, where elements have finite support. This is a definition, not an assertion that every representation has such a decomposition. The zero representation is the empty direct sum and is therefore completely reducible under this convention.

A representation $ρ : g \to g l (V)$ is faithful if $ρ$ is injective.

Irreducible, completely reducible, and faithful representations

Definition

Depends on

Dependency tree · 2 levels

Sources